Discrete Optimization Problems for Radiation Therapy Planning

A well-studied problem in intensity modulated radiation therapy (IMRT) is the representation of a given intensity matrix, i.e. a matrix of nonnegative integers, as a nonnegative linear combination of special 0-1-matrices, called segments. These segments can be practically realized by multileaf collimators (MLC). One important aim is the minimization of the sum of the coefficients of the linear combination, i.e. the delivery time. This paper gives an introduction into this subject and surveys recent results for related problems that arise in practical applications. In particular, segmentation problems are discussed where not exactly the given matrix but a matrix with small deviations has to be segmented. Moreover, restrictions on the segments like the rectangular constraint, the interleaf collision constraint and the tongue-andgroove constraint are considered.